A couple simple linear algebra questions.

[charlies1902]

I think I know the answer to these questions, but I just want to make sure.

1) If A is invertible then A+A is invertible. True/False
True.

Because det(A)≠0, det(A+A)=det(2A)=2^n * det(A)≠0

Is this correct.

2) A 3x3 matrix can have 2 distinct eigenvalues. True/False
True, although I was kind of confused with what "distinct" means.
The characteristic polynomial can look something like this: (λ-1)^2 * (λ+2)
Distinct just refers to the # of "different" eigenvalues right? And doesn't include them again if they're repeated?

 

[STEMucator]

Yup, you first one looks good. As does your second one.

Distinct does indeed refer to the amount of DIFFERENT eigenvalues a matrix has in this case.

In general, if you have an nxn matrix, it CAN have as many as n distinct eigenvalues.

 

[charlies1902]

In general, if you have an nxn matrix, it CAN have as many as n distinct eigenvalues.

so basically 0≤λ≤n?

 

[STEMucator]

so basically 0≤λ≤n?

What does this statement mean? What is λ?

I think what you're intending is to say that there are going to be greater than or equal to zero eigenvalues for a nxn matrix, this is incorrect to assume. Is a matrix really a matrix if n = 0? Even the statement makes me cringe.

Really, we're only concerned with the eigenvalues of matrices for n ≥ 2 and only square matrices are considered.

So supposing that A is an nxn matrix. The fewest eigenvalues it can have is 1 and that's only if the algebraic multiplicity of the eigenvalue equals n. Otherwise it can have up to n eigenvalues. I hope that clears that up for you :).